Finite Group Databasebeta

← #10590604

Subgroups — #10590604

Order 576.

317 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 317 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×Dic6C211yesmaximal, characteristic
2C3×#897C211yesmaximal, characteristic
3C2×#3719C211yesmaximal, characteristic, complemented
4C2×#3718C211yesmaximal, characteristic, complemented
5#63637C211yesmaximal, characteristic, complemented
6C3×#810C211yesmaximal, characteristic, complemented
7(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic
8#5050—31nomaximal
9#4809—31nomaximal
10C2×#634C2211yescharacteristic
11C2×C3×C4×D3C2211yescharacteristic
12C2×C3×Dic6C2211yescharacteristic
13C2×C3×Dic6C2211yescharacteristic
14C2×#627—21no—
15C2×D3×Dic3—21no—
16D3×Dic6—22no—
17D3×Dic6—22no—
18C32×C4⋊C4C2211yescharacteristic
19C2×C3⋊(C3⋊C8)C2211yescharacteristic
20C3×(C2×D3)⋊C4—21no—
21C3×C4⋊Dic3—21no—
22C2×C3×C3⋊C8C2211yescharacteristic
23#3719—22no—
24#3718—22no—
25C2×D3×Q8—31no—
26C22×Dic6—31no—
27C3×C22⋊Q8—31no—
28C2×#216—31no—
29C2×#216—31no—
30C2×#216—31no—
31C2×Dic12—31no—
32#897D311yescharacteristic
33C3×#103—31no—
34#830—31no—
35(C2×D3)⋊C8—31no—
36#810—31no—
37#810—31no—
38#810—61no—
39#810D311yescharacteristic, complemented
40C22×C3×D3D411yescharacteristic
41C2×C32×C4—21no—
42C2×C32×C4C2311yescharacteristic
43C2×C3⋊Dic3—21no—
44#634—21no—
45#634D412yes—
46C2×C3×Dic3—21no—
47C2×C3×Dic3—21no—
48C2×C3×Dic3D411yescharacteristic
49C3×C4×D3—21no—
50C3×C4×D3—21no—
51C3×Dic6—21no—
52C3×Dic6—21no—
53C3×Dic6—21no—
54C3×Dic6D412yes—
55#627—24no—
56D3×Dic3—24no—
57C3⋊(C3⋊C8)—21no—
58C3×C3⋊C8—21no—
59#451—91noSylow
60C2×C3×Q8—31no—
61C2×C3×Q8—31no—
62C22×C3×C4—31no—
63C22×Dic3—61no—
64D3×Q8—62no—
65D3×Q8—62no—
66C2×C4×D3—61no—
67C2×C4×D3C2×D311yescharacteristic
68C2×Dic6—31no—
69C2×Dic6—31no—
70C2×Dic6—61no—
71C2×Dic6—62no—
72C2×Dic6—61no—
73C2×Dic6—61no—
74C2×Dic6—62no—
75C2×Dic6C2×D311yescharacteristic
76C2×Dic6C2×D311yescharacteristic
77C2×C3×C8—31no—
78C3×C4⋊C4—21no—
79C3×C4⋊C4—61no—
80C3×C4⋊C4C2×D311yescharacteristic
81C3×C4⋊C4C2×D311yescharacteristic
82C3×C22⋊C4—61no—
83#216—62no—
84#216—62no—
85#216—62no—
86(C2×D3)⋊C4—21no—
87C4⋊Dic3—21no—
88C2×C3⋊C8—31no—
89C2×C3⋊C8—31no—
90C2×C3⋊C8—61no—
91C2×C3⋊C8C2×D311yescharacteristic
92Dic12—62no—
93C22×C32C2×D411yescharacteristic
94C2×C3×D3—21no—
95C2×C3×D3—21no—
96C2×C3×D3Q1612yescomplemented
97C32×C4—41no—
98C32×C4C2×D411yescharacteristic
99C32×C4C2×D411yescharacteristic
100C3⋊Dic3—22no—
101C3×Dic3—22no—
102C3×Dic3—21no—
103C3×Dic3—41no—
104C22×Q8—91no—
105C2×Q16—91no—
106C2×Q16—91no—
107C22⋊Q8—31no—
108#103—31no—
109#103—91no—
110C22⋊C8—91no—
111C23×C3—31no—
112C22×D3#6711yescharacteristic
113C3×Q8—32no—
114C3×Q8—61no—
115C3×Q8—61no—
116C3×Q8—61no—
117C2×C3×C4—21no—
118C2×C3×C4—21no—
119C2×C3×C4—21no—
120C2×C3×C4—31no—
121C2×C3×C4—41no—
122C2×C3×C4—61no—
123C2×C3×C4—61no—
124C2×C3×C4—61no—
125C2×C3×C4—61no—
126C2×C3×C4C22×D311yescharacteristic
127C2×C3×C4C22×D311yescharacteristic
128C2×Dic3—21no—
129C2×Dic3—21no—
130C2×Dic3—64no—
131C2×Dic3—61no—
132C2×Dic3—61no—
133C2×Dic3—121no—
134C2×Dic3#6711yescharacteristic
135C4×D3—21no—
136C4×D3—21no—
137C4×D3—64no—
138Dic6—21no—
139Dic6—21no—
140Dic6—21no—
141Dic6—32no—
142Dic6—32no—
143Dic6—62no—
144Dic6—64no—
145Dic6—64no—
146Dic6—61no—
147Dic6—61no—
148Dic6—121no—
149Dic6D1212yes—
150C3×C8—61no—
151C3⋊C8—21no—
152C3⋊C8—61no—
153C3⋊C8—61no—
154C3⋊C8—121no—
155C2×C32#12011yescharacteristic
156C2×C32C2×Q1611yescharacteristic
157C2×C32#13711yescharacteristic
158C3×D3—22no—
159C2×Q8—31no—
160C2×Q8—31no—
161C2×Q8—91no—
162C2×Q8—181no—
163C2×Q8—182no—
164C2×Q8—182no—
165C22×C4—31no—
166C22×C4—181no—
167Q16—182no—
168Q16—182no—
169C2×C8—31no—
170C2×C8—91no—
171C4⋊C4—61no—
172C4⋊C4D3211yescharacteristic
173C22⋊C4—61no—
174C22×C3—21no—
175C22×C3—32no—
176C22×C3—61no—
177C22×C3—61no—
178C22×C3D3×D411yescharacteristic
179C22×C3C2×#6711yescharacteristic
180C2×D3—21no—
181C2×D3—21no—
182C2×D3#21612yescomplemented
183C3×C4—21no—
184C3×C4—21no—
185C3×C4—41no—
186C3×C4—41no—
187C3×C4—61no—
188C3×C4—62no—
189C3×C4—81no—
190C3×C4—121no—
191C3×C4C2×D1211yescharacteristic
192C3×C4D3×D411yescharacteristic
193C3×C4D3×D411yescharacteristic
194C3×C4D3×D411yescharacteristic
195Dic3—22no—
196Dic3—21no—
197Dic3—41no—
198Dic3—62no—
199Dic3—62no—
200Dic3—122no—
201C32#45111yescharacteristic, complemented, Sylow
202C23—31no—
203Q8—32no—
204Q8—61no—
205Q8—61no—
206Q8—61no—
207Q8—92no—
208Q8—184no—
209Q8—181no—
210C2×C4—21no—
211C2×C4—31no—
212C2×C4—61no—
213C2×C4—61no—
214C2×C4—61no—
215C2×C4—61no—
216C2×C4—184no—
217C2×C4—181no—
218C2×C4C2×D3211yescharacteristic
219C8—61no—
220C8—181no—
221C2×C3—21no—
222C2×C3—21no—
223C2×C3—21no—
224C2×C3—62no—
225C2×C3#88211yescharacteristic
226C2×C3#91511yescharacteristic
227C2×C3D3×Q1611yescharacteristic
228C2×C3#93711yescharacteristic
229C2×C3C2×#21611yescharacteristic
230C2×C3#95111yescharacteristic
231D3—22no—
232C22—32no—
233C22—61no—
234C22—61no—
235C22D3×#6711yescharacteristic
236C4—41no—
237C4—61no—
238C4—62no—
239C4—121no—
240C4—182no—
241C4D3×D1211yescharacteristic
242C4#380211yescharacteristic
243C3—21no—
244C3#480911yescharacteristic, complemented
245C3#505011yescharacteristic, complemented
246C2—62no—
247C2#6392211yescharacteristic
248C2#6392911yescharacteristic
249C2D3×#21611yescharacteristic
subgroupG/H
centerC22D3×#67
commutator, cosocleC2×C32×C4C23
FrattiniC2×C4C2×D32
FittingC32×C4⋊C4C22
socleC22×C32C2×D4
prime radicalC22×C3×D3D4
hypercenterC4⋊C4D32
hypocenterC32#451
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×C4 ◁ C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. C2×#3719C2semidirect
4. C2×#3718C2semidirect
5. #63637C2semidirect
6. C3×#810C2semidirect
39. #810D3semidirect
96. C2×C3×D3Q16semidirect
182. C2×D3#216semidirect
201. C32#451semidirect
244. C3#4809semidirect
245. C3#5050semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×D3×Dic62
C3×#8972
C2×#37192
C2×#37182
#636372
C3×#8102
(C2×D3)⋊(C3⋊C8)2
#50503
#48093